JEE Main 2024 Mathematics question
Let A be a non-singular matrix of order 3. If det(3adj(2adj((det A)A))) = 3⁻¹³·2⁻¹⁰ and det(3adj(2A)) = 2ᵐ·3ⁿ, then |3m + 2n| is equal to .......
Answer: 14
Solution
For a 3 × 3 matrix, det(cM) = c³det M and adj(adj M) = (det M)M. Let d = det A. Then adj(2 adj(dA)) = 4d⁴ adj(adj A) = 4d⁵A, so det(3adj(2adj(dA))) = det(12d⁵A) = 12³d¹⁶ = 2⁶3³d¹⁶. Setting this equal to 2⁻¹⁰3⁻¹³ gives
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